The Nine Canticles are nine songs from the Bible that are usually appended to Orthodox Psalters. They are theoretically chanted at Matins (though they have long fallen out of use, except perhaps in monasteries). The nine odes of canons are based on these nine odes/canticles.

As Rufus said, the Canticles are the thematic basis of all canons (although the newer the canons are, the less they tend to reflect the Canticles).

As I understand it, the Canticles are still to be sung at daily matins during the Triodion period (the three Canticles for which there are odes...hence the name "Triodion"). Not sure how much it's done, as most parishes don't have daily matins at all, much less follow the rubrics perfectly. You're unlikely to hear the Canticles sung outside of a monastery, at least in the West.

The Rite of the 12 Psalms is generally used as a replacement for other services, sung when there is no priest available, or when all the required service rubrics (menaion, octoechoes, horologion, etc.) are either not available or too difficult to use.

I have a PDF of a Russian psalter on my computer, which has these notes:

The Rite for Singing the Twelve Psalms: which the venerable fathers of the desert were wont to sing day and night; concerning which, account is found in the books of the fathers and in the lives and sufferings of many saints. This rite was brought to Russia from the Holy Mountain by Dositheus, Archimandrite of the Kiev Caves.

The Rite of the Twelve Psalms is not a form of cell rule, but a substitute for the canonical hours (Vespers, Matins, etc.) when it is not possible to conduct the regular services. The cell rule should still be observed, even when the Rite of the Twelve Psalms is performed. This rite can be used in sketes where there is limited energy or resources for full services, it can be used while traveling, or by those who simply wish to have a rule of prayer unencumbered by many books.

During the Holy Quadragesima: In place of the Hours and Vespers, we sing nine psalms. In place of Great Compline, the remaining three. But at night, all 12 psalms, and the dismissal...

Outside of the Holy Quadragesima: In place of the Hours, we sing the first six psalms. In place of Vespers and Compline, the last six. But in the night (i.e., for Matins), we sing 12 psalms, and we perform the Lesser Dismissal...

The Rite could probably find a place among pious laypeople whose parish either does not offer daily services, or is too far away to attend.

For those who do not have a Psalter with this rite in it (and most do not have it), it consists of:

The usual beginningPsalms 26, 31, 56Trisagion and TropariaPsalms 33, 38, 40Trisagion and TropariaPsalms 69, 70, 76Trisagion and TropariaPsalm 101, Prayer of Manasseh, Small DoxologyPrayer of St. EustratiusTrisagion and TropariaDismissal